Linking Number

A Link invariant. Given a two-component oriented Link, take the sum of $+1$ crossings and $-1$ crossing over all crossings between the two links and divide by 2. For components $\alpha$ and $\beta$,

$$L(\alpha,\beta)\equiv {\textstyle{1\over 2}}\sum_{p\in \alpha\sqcap\beta} \epsilon(p),$$

where $\alpha\sqcap\beta$ is the set of crossings of $\alpha$ with $\beta$ and $\epsilon(p)$ is the sign of the crossing. The linking number of a splittable two-component link is always 0.

See also Jones Polynomial, Link

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 132-133, 1976.